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Generalized Pauli matrices
In and , in particular , the term .}} Here, a few classes of such matrices are summarized. Pauli matrices : \begin{align} \sigma_1 = \sigma_x &= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \\ \sigma_2 = \sigma_y &= \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \\ \sigma_3 = \sigma_z &= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix} \,. \end{align} Generalized Gell-Mann matrices (Hermitian) Construction Let be the matrix with 1 in the -th entry and 0 elsewhere. Consider the space of d''×''d complex matrices, , for a fixed d. Define the following matrices, : }} :: , for . :: , for . : }} :: , the identity matrix, for 1}},. :: , for . :: \sqrt{\tfrac{2}{d(d-1)}} \left( h_1^{d-1} \oplus (1 - d)\right) = \sqrt{\tfrac{2}{d(d-1)}} \left( I_{d-1} \oplus (1 - d)\right), for d''}}. The collection of matrices defined above without the identity matrix are called the ''generalized Gell-Mann matrices, in dimension . The symbol ⊕ (utilized in the above) means . The generalized Gell-Mann matrices are and by construction, just like the Pauli matrices. One can also check that they are orthogonal in the on . By dimension count, one sees that they span the vector space of complex matrices, \mathfrak{gl} ( ,ℂ). They then provide a Lie-algebra-generator basis acting on the fundamental representation of \mathfrak{su} ( ). In dimensions = 2 and 3, the above construction recovers the Pauli and , respectively. : and g_i = \lambda_i/2 . A non-Hermitian generalization of Pauli matrices The Pauli matrices \sigma _1 and \sigma _3 satisfy the following: : \sigma _1 ^2 = \sigma _3 ^2 = I, \; \sigma _1 \sigma _3 = - \sigma _3 \sigma _1 = e^{\pi i} \sigma _3 \sigma_1. The so-called is : W = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}. Like the Pauli matrices, ''W is both and . \sigma _1, \; \sigma _3 and W'' satisfy the relation : \; \sigma _1 = W \sigma _3 W^* . The goal now is to extend the above to higher dimensions, ''d, a problem solved by (1882). : \begin{align} H_1 &= \begin{bmatrix} 1 \end{bmatrix}, \\ H_2 &= \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, \\ H_4 &= \begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 \end{bmatrix}, \end{align} and : H_{2^k} = \begin{bmatrix} H_{2^{k-1}} & H_{2^{k-1}}\\ H_{2^{k-1}} & -H_{2^{k-1}} \end{bmatrix} = H_2 \otimes H_{2^{k-1}}, Sylvester's matrices have a number of special properties. They are and, when k'' ≥ 1, have zero. The elements in the first column and the first row are all . The elements in all the other rows and columns are evenly divided between . Sylvester matrices are closely connected with s. Construction: The clock and shift matrices Fix the dimension as before. Let exp(2''πi''/''d'')}}, a root of unity. Since 1}} and , the sum of all roots annuls: : 1 + \omega + \cdots + \omega ^{d-1} = 0 . Integer indices may then be cyclically identified mod . Now define, with Sylvester, the shift matrix : \Sigma _1 = \begin{bmatrix} 0 & 0 & 0 & \cdots &0 & 1\\ 1 & 0 & 0 & \cdots & 0 & 0\\ 0 & 1 & 0 & \cdots & 0 & 0\\ 0 & 0 & 1 & \cdots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots &\vdots &\vdots \\ 0 & 0 &0 & \cdots & 1 & 0\\ \end{bmatrix} and the clock matrix, : \Sigma _3 = \begin{bmatrix} 1 & 0 & 0 & \cdots & 0\\ 0 & \omega & 0 & \cdots & 0\\ 0 & 0 &\omega ^2 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & 0 & \cdots & \omega ^{d-1} \end{bmatrix}. These matrices generalize σ''1 and ''σ''3, respectively. Note that the unitarity and tracelessness of the two Pauli matrices is preserved, but not Hermiticity in dimensions higher than two. Since Pauli matrices describe , Sylvester dubbed the higher-dimensional analogs "nonions", "sedenions", etc. These two matrices are also the cornerstone of '''quantum mechanical dynamics in finite-dimensional vector spaces' as formulated by , and find routine applications in numerous areas of mathematical physics. They are (finite-dimensional) representations of the corresponding elements of the on a d''-dimensional Hilbert space. The following relations echo and generalize those of the Pauli matrices: : \Sigma _ 1 ^d = \Sigma _ 3 ^d = I and the braiding relation, : \; \Sigma_3 \Sigma _1 = \omega \Sigma_1 \Sigma _3 = e^{2 \pi i / d} \Sigma_1 \Sigma _3 , the , and can be rewritten as : \; \Sigma_3 \Sigma _1 \Sigma _3^{d-1} \Sigma_1 ^{d-1} = \omega ~. On the other hand, to generalize the Walsh–Hadamard matrix ''W, note : W = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & \omega ^{2 -1} \end{bmatrix} = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \\ 1 & \omega ^{d -1} \end{bmatrix}. Define, again with Sylvester, the following analog matrix, still denoted by W'' in a slight abuse of notation, : W = \frac{1}{\sqrt{d}} \begin{bmatrix} 1 & 1 & 1 & \cdots & 1\\ 1 & \omega^{d-1} & \omega^{2(d-1)} & \cdots & \omega^{(d-1)^2}\\ 1 & \omega^{d-2} & \omega^{2(d-2)} & \cdots & \omega^{(d-1)(d-2)}\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 &\omega &\omega ^2 & \cdots & \omega^{d-1} \end{bmatrix}~. It is evident that ''W is no longer Hermitian, but is still unitary. Direct calculation yields : \; \Sigma_1 = W \Sigma_3 W^* ~, which is the desired analog result. Thus, , a , arrays the eigenvectors of , which has the same eigenvalues as . , converting position coordinates to momentum coordinates and vice versa.}} The complete family of ''d''2 unitary (but non-Hermitian) independent matrices provides Sylvester's well-known trace-orthogonal basis for \mathfrak{gl} (''d,ℂ), known as "nonions" \mathfrak{gl} (3,ℂ), "sedenions" \mathfrak{gl} (4,ℂ), etc... This basis can be systematically connected to the above Hermitian basis. (For instance, the powers of , the , map to linear combinations of the s.) It can further be used to identify \mathfrak{gl} (d'',ℂ) , as , with the algebra of . References Category:Advanced mathematics